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Abstract

Let I be an ideal of the ring of Laurent polynomials K [x ±11 , . . . , x ±1 n ] with coefficients in a real-valued field (K, v). The fundamental theorem of tropical algebraic geometry states the equality trop(V (I)) = V (trop(I)) between the tropicalization trop(V (I)) of the closed subscheme V (I) ⊂ (K * ) n and the tropical variety V (trop(I)) associated to the tropicalization of the ideal trop(I).In this work we prove an analogous result for a differential ideal G of the ring of differential polynomials K [[t]]{x 1 , . . . , x n }, where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) ⊂ K [[t]] n of G, and the set of solutions Sol(trop(G)) ⊂ P(‫ޚ‬ ≥0 ) n associated to the tropicalization of the ideal trop(G). These two sets are linked by a tropicalization morphism trop : Sol(G) → Sol(trop(G)).We show the equality trop(Sol(G)) = Sol(trop(G)), answering a question recently raised by D. Grigoriev.